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Practice problems A lunar excursion module has a mass of 2000 kg and is supported by four symmetrically located legs, each of which can be approximated as a springdamper system with negligible mass. Design the springs and dampers of the system in order to have the damped period of vibration between 1 s and 2 s.

Design PHYSICAL DYNAMIC SYSTEM

To design K, C so that T is between 1 sec to 2 sec.

VIBRATORY MODEL

GOVERNING EQUATIONS

SOLVE GOVERNING EQUATIONS

K

INTERPRETE RESULTS

Ke α

K e = K cos2 α

Theory m!! x + cx! + kx = f (t)

m!! x + cx! + kx = 0

2π T= ωd

ωd = ωn c ξ= cc

(1− ξ ) 2

Ke ωn = m

cc = 2mω n

Hoisting Drum

Find the natural frequency of the system. Comment on the parameters of the system. Is the system nonlinear?

A daredevil motorcyclist jumps up and lands back on the ground from a height of 4m. Assume that the mass of the rider and motorcycle is 300 kg and the stiffness of the suspension is 5 kN/m and damping ratio of 0.5. Find the subsequent oscillatory motion of the motorcycle and plot the motion for 5 sec. Assume that the system is modeled as SDOF system and both wheels of the motorcycle touch the ground at the same time and the motorcycle does not bounce back after landing.

https://goo.gl/images/ErLcRJ

Figure 5 shows a spacecraft with four solar panels. Each panel has the dimensions 150cm × 90cm × 2cm. with a weight density of 3 gm/cm3 , and is connected to body of the spacecraft by aluminum rods of length 30 cm and diameter 2 cm. Assuming that the body of the spacecraft is very large (rigid), determine the natural frequency of vibrations of each panel. How can this information be useful to the designer?

An overhead traveling crane

36

Damping Model • Viscous Damping – System vibrates in viscous medium. Viscous dissipative force is modeled as: F = cx! F = cx! d

d

Cyclic Energy dissipation: Wd =π cω X 2

Source: http://www.myrepurposedlife.com/wpcontent/uploads/2014/02/storm-door-closer.jpg

http://taylordevices.com/papers/history/design.htm

x

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Damping Model • Hysteretic /Material /Structural Damping – Material is cyclically stressed; energy dissipated due to intermolecular friction – Valid for harmonic excitation. Energy dissipation Wd =α X 2 α c = ceq by comparing energies: e q πω Combining damping & stiffness: !" = $ 1 + '( )* +,-

ΔW = πβ KX

2

α FR = x! + Kx πω

K(1+ j β ) is Complex Stiffness and

β=

α is structural damping factor or loss factor πK 40

Damping Model

(contd..)

Dry Friction or Coulomb Damping Damping force is assumed constant and oppose the direction of motion. The decay of free vibration is linear rather than exponential as seen in viscous damping.

Wd =4µ WX

Ce =

4µW πω X

! Fd = µWsign( x) x

Free Vibration with Coulomb Damping

Case (a)

Case (b)

(a)

(b)

The motion stops when No.of half cycles r after which the motion ceases:

The slope of the enveloping straight line

1. The equation of motion is nonlinear with Coulomb damping, while it is linear with viscous damping. 2. The natural frequency of the system is unaltered with the addition of Coulomb damping, while it is reduced with the addition of viscous damping. 3. The motion is periodic with Coulomb damping, while it can be nonperiodic in a viscously damped (overdamped) system. 4. The system comes to rest after some time with Coulomb damping, whereas the motion theoretically continues forever (perhaps with an infinitesimally small amplitude) with viscous and hysteresis damping. 5. The amplitude reduces linearly with Coulomb damping, whereas it reduces exponentially with viscous damping. 6. In each successive cycle, the amplitude of motion is reduced by the amount so the amplitudes at the end of any two consecutive cycles are related:

Viscous Vs. Coulomb • Distinction based on – Frequency of oscillations – Nature of governing equations – Nature of decay of oscillation amplitude – Final amplitude as t -> ∞ – Damping force: constant vs. velocity dependent – Motion: strictly periodic vs. aperiodic